OPERATIONS SCHEDULING SUPPLEMENT J J-1 J PERATIONS …€¦ · J-2 SUPPLEMENT J OPERATIONS...

OPERATIONS SCHEDULING SUPPLEMENT J J-1

LEARNING GOALSAfter reading this supplement, you shouldbe able to:

1. Define new performance measures (beyondflow time and past due) for evaluating aschedule.

2. Describe the decision rules (beyond FCFSand EDD) to sequence jobs.

3. Determine schedules for single andmultiple workstations.

T his supplement focuses on operationsscheduling, which involves assigning jobs toworkstations or employees to jobs for speci-

fied time periods. Effective scheduling helps man-agers achieve the full potential of their supplychains. Chapter 14, “Operations Planning andScheduling,” covers the basics of scheduling—Ganttcharts, workforce scheduling, two rules (FCFS andEDD) for sequencing work at a single workstation,and two commonly used performance measures(flow time and past due). Here we deepen yourunderstanding with additional performance mea-sures and priority sequencing rules, a discussion ofscheduling multiple workstations, and a discussion ofscheduling a two-station flow shop.

OPERATIONS SCHEDULINGJ

myomlab and the Companion Website atwww.pearsonhighered.com contain many tools,activities, and resources designed for this supplement.

J-2 SUPPLEMENT J OPERATIONS SCHEDULING

Scheduling Service and ManufacturingProcessesThe scheduling techniques we discuss in this supplement cut across the various processtypes found in services and manufacturing. Many service firms are characterized by afront-office process with high customer contact, divergent work flows, customization,and, consequently, a complex scheduling environment. Often customer demands aredifficult to predict, which puts a high premium on scheduling employees to handle thevaried needs of customers. At the other extreme in the service industry, a back-officeprocess has low customer involvement, uses more line work flows, and provides stan-dardized services. Inanimate objects are processed; these processes take on the appear-ance of manufacturing processes.

Manufacturing processes also benefit from operations scheduling techniques. Ourdiscussion of the operations scheduling techniques in this supplement has applicationfor job, batch, and line processes in services as well as in manufacturing. Schedules forcontinuous processes can be developed with linear programming (see Supplement E,“Linear Programming”). Although the scheduling techniques in this chapter providesome structure to the selection of good schedules, many alternatives typically need to beevaluated. We begin by looking at the performance measures managers use to selectgood schedules.

Performance MeasuresWe already covered two important performance measures in Chapter 14, “OperationsPlanning and Scheduling.” Flow time is the time a job spends in the service or manufactur-ing system, and past due (tardiness) is the amount of time by which a job missed its duedate. In this regard, a job is the object receiving service or being manufactured. For example,a job may be a customer waiting for service at a state licensing bureau or it may be a batchof pistons waiting for a manufacturing process. These two performance measures can beinsufficient, depending on the competitive priorities of a process. Additional performancemeasures follow:

� Makespan. The total amount of time required to complete a group of jobs is calledmakespan. Minimizing makespan supports the competitive priorities of cost (lowerinventory) and time (delivery speed).

� Total Inventory. This performance measure is used to measure the effectivness ofschedules for manufacturing processes. The sum of scheduled receipts and on-handinventories is the total inventory.

Minimizing total inventory supports the competitive priority of cost (inventory hold-ing costs).

� Utilization. The degree to which equipment, space, or the workforce is currently beingused, measured as the ratio of the average output rate to maximum capacity.Maximizing the utilization of a process supports the competitive priority of cost (slackcapacity).

These performance measures often are interrelated. For example, minimizingthe average flow time tends to increase utilization. Minimizing the makespan for a groupof jobs tends to increase utilization. Understanding how flow time, makespan, past due,and utilization interact can make the selection of good schedules easier.

Sequencing JobsOperations schedules are short-term plans designed to implement the sales and opera-tions plan. Often, several jobs must be processed at one or more workstations. Typically,a variety of tasks can be performed at each workstation. If schedules are not carefullyplanned to avoid bottlenecks, waiting lines may develop. For example, Figure J.1 depicts

Total inventory = Scheduled receipts for all items + On-hand inventories of all items

Makespan = Time of completion of last job - Starting time of the first job

operations scheduling

A type of scheduling in which jobs areassigned to workstations or employeesare assigned to jobs for specified timeperiods.

makespan

The total amount of time required tocomplete a group of jobs.

total inventory

The sum of scheduled receipts and on-hand inventories.


the complexity of scheduling a manufacturing process. When a job order is received fora part, the raw materials are collected and the batch is moved to its first operation. Thecolored arrows show that jobs follow different routes through the manufacturingprocess, depending on the product being made. At each workstation, the next job toprocess is a decision because the arrival rate of jobs at a workstation often differs fromthe processing rate of the jobs at a workstation, thereby creating a waiting line. In addi-tion, new jobs can enter the process at any time, thereby creating a dynamic environ-ment. Such complexity puts pressure on managers to develop scheduling proceduresthat will handle the workload efficiently.

In this section, we focus on scheduling approaches used in two environments: (1) diver-gent flow processes and (2) line flow processes. A manufacturer's operation with divergentflows is often called a job shop, which specializes in low- to medium-volume productionand utilizes job or batch processes. The front office would be the equivalent for a serviceprovider. Jobs in divergent flow processes are difficult to schedule because of the variabilityin job routings and the continual introduction of new jobs to be processed. Figure J.1 depictsa manufacturer’s job shop. A manufacturer's operation with line flows is often called a flowshop, which specializes in medium- to high-volume production and utilizes line orcontinuous flow processes. The back office would be the equivalent for a service provider.Tasks are easier to schedule because the jobs have a common flow pattern through the sys-tem. Nonetheless, scheduling mistakes can be costly in either situation.

Job Shop SequencingJust as many schedules are feasible for a specific group of jobs at a particular set of workstations,numerous methods can be used to generate schedules. They range from straightforwardmanual methods, such as manipulating Gantt charts, to sophisticated computer models fordeveloping optimal schedules. One way to generate schedules in job shops is by usingpriority sequencing rules, which allows the schedule for a workstation to evolve over a periodof time. The decision about which job to process next is made with simple priority ruleswhenever the workstation becomes available for further processing. One advantage of thismethod is that last-minute information on operating conditions can be incorporated intothe schedule as it evolves.

We already covered two important sequencing rules in Chapter 14, “Operations Planningand Scheduling.” The first-come, first-served (FCFS) rule gives the job arriving at the workstationfirst the highest priority. The earliest due date (EDD) rule gives the job with the earliest due datebased on assigned due dates the highest priority. Such rules can be applied by a worker or

job shop

A manufacturer's operation that specializesin low- to medium-volume production andutilizes job or batch processes.

flow shop

A manufacturer's operation thatspecializes in medium- to high-volumeproduction and utilizes line or continuousflow processes.

� FIGURE J.1Diagram of a Manufacturing JobShop Process

Raw

mat

eria

ls

Ship

ping

dep

artm

ent

Legend:

Batch of parts

Workstation


incorporated into a computerized scheduling system that generates a dispatch list of jobs andpriorities for each workstation. Additional priority sequencing rules follow:

� Critical Ratio. The critical ratio (CR) is calculated by dividing the time remaining untila job’s due date by the total shop time remaining for the job, which is defined as thesetup, processing, move, and expected waiting times of all remaining operations,including the operation being scheduled. The formula is

critical ratio (CR)

A ratio that is calculated by dividing thetime remaining until a job’s due date by thetotal shop time remaining for the job, whichis defined as the setup, processing, move,and expected waiting times of all remainingoperations, including the operation beingscheduled.

shortest processing time (SPT)

A priority sequencing rule that specifies thatthe job requiring the shortest processingtime is the next job to be processed.

slack per remaining operations (S/RO)

A priority sequencing rule thatdetermines priority by dividing the slackby the number of operations that remain,including the one being scheduled.

single-dimension rules

A set of rules that bases the priority of ajob on a single aspect of the job, such asarrival time at the workstation, the duedate, or the processing time.

Comparing the EDD and SPT RulesEXAMPLE J.1

The Taylor Machine Shop rebores engine blocks. Currently, five engine blocks are waiting for processing. At anytime, the company has only one engine expert on duty who can do this type of work. The engine problems havebeen diagnosed, and the processing times for the jobs have been estimated. Expected completion times havebeen agreed upon with the shop’s customers. The accompanying table shows the current situation. Because the

Tutor J.1 in myomlab provides a newexample to practice EDD and SPT rules.

The difference between the due date and today’s date must be in the same time units asthe total shop time remaining. A ratio less than 1.0 implies that the job is behind sched-ule, and a ratio greater than 1.0 implies that the job is ahead of schedule. The job withthe lowest CR is scheduled next.

� Shortest Processing Time. The job requiring the shortest processing time (SPT) at theworkstation is processed next.

� Slack per Remaining Operations. Slack is the difference between the time remaininguntil a job’s due date and the total shop time remaining, including that of the operationbeing scheduled. A job’s priority is determined by dividing the slack by the number ofoperations that remain, including the one being scheduled, to arrive at the slack perremaining operations (S/RO).

The job with the lowest S/RO is scheduled next. Ties are broken in a variety of ways iftwo or more jobs have the same priority. One way is to arbitrarily choose one of the tiedjobs for processing next.

Although the priority sequencing rules seem simple, the actual task of scheduling hun-dreds of jobs through hundreds of workstations requires intensive data gathering and manip-ulation. The scheduler needs information on each job’s processing requirements: the job’sdue date; its routing; the standard setup, processing, and expected waiting times at eachoperation; whether alternative workstations could be used at each operation; and the inputsfrom internal or external suppliers at each operation. In addition, the scheduler needs toknow the job’s current status: its location (waiting in line for a workstation or being processedat a workstation), how much of the operation has been completed, the actual arrival anddeparture times at each operation or waiting line, and the actual processing and setup times.The scheduler or software uses the priority sequencing rules to determine the processingsequence of jobs at a workstation and the remaining information for estimating job arrivaltimes at the next workstation, as well as determining whether an alternative workstationshould be used when the primary one is busy. Because this information may change through-out the day, computers are needed to track the data and to maintain valid priorities.

Sequencing Jobs for One WorkstationAny priority sequencing rule can be used to schedule any number of workstations. Forthe purpose of illustrating the rules, however, we focus on scheduling several jobs at asingle workstation. We divide the rules into two categories: (1) single-dimension rulesand (2) multiple-dimension rules.

Single-Dimension Rules Some priority sequencing rules (e.g., FCFS, EDD, and SPT)base a job’s priority assignment only on information about the jobs waiting for processing atthe individual workstation. We call these rules single-dimension rules because they deter-mine priority based on a single aspect of the job, such as arrival time at the workstation, thedue date, or the processing time. We begin with an example of single-dimension rules.

S/RO =

(Due date - Today’s date) - Total shop time remaining

Number of operations remaining

CR =

Due date - Today’s date

Total shop time remaining


Taylor Machine Shop is open from 8:00 A.M. until 5:00 P.M. each weekday, plus weekend hours as needed, thecustomer pickup times are measured in business hours from the current time. Determine the schedule for theengine expert by using (a) the EDD rule and (b) the SPT rule. For each rule, calculate the average flow time, aver-age hours early, and average hours past due. If average past due is most important, which rule should be chosen?

SOLUTION

a. The EDD rule states that the first engine block in the sequence is the one with the closest due date.Consequently, the Ranger engine block is processed first. The Thunderbird engine block, with its duedate furthest in the future, is processed last. The sequence is shown in the following table, along withthe flow times, the hours early, and the hours past due.

The flow time for each job is its finish time, plus the time since the job arrived.1 For example, the Explorerengine block’s finish time will be 14 hours from now (8 hours waiting time before the engine expert startedto work on it plus 6 hours processing). Adding the 10 hours since the order arrived at this workstation(before the processing of this group of orders began) results in a flow time of 24 hours. You might think ofthe sum of flow times as the total job hours spent by the engine blocks since their orders arrived at theworkstation until they were processed.

The performance measures for the EDD schedule for the five engine blocks are

Average hours past due =

0 + 2 + 0 + 12 + 225

= 7.2 hrs

Average hours early =

2 + 0 + 1 + 0 + 05

= 0.6 hrs

Average flow time =

20 + 24 + 20 + 33 + 445

= 28.2 hrs

1 Flow time, as a performance measure in its traditional use, does not count the time a job spends “out-side the system under our control.” Our “system” in this supplement is the single workstation (or twoworkstations in the case of Johnson’s rule in the next section). Arrival time here relates to when the jobwas first available for processing at the workstation. Adding the time since the order arrived at the work-station to the job’s finish time departs from conventions used in early research on static problems,which assumed that no jobs arrive during the time span covered by the resulting schedule. With tradi-tional assumptions, a job’s finish time and flow time are identical and SPT will always have the best flowtime performance. With our definition of flow time, the SPT rules do not necessarily produce the bestflow time performance, such as when the job with the shortest processing time arrived at the workstationwell before the other jobs.

Engine BlockBusiness Hours

Since Order ArrivedProcessing Time,

Including Setup (hours)

Business Hours Until Due Date

(customer pickup time)

Ranger 12 8 10

Explorer 10 6 12

Bronco 1 15 20

Econoline 150 3 3 18

Thunderbird 0 12 22

Engine BlockSequence

HoursSince Order

ArrivedBeginWork

ProcessingTime (hr)

FinishTime (hr)

Flow Time(hr)

ScheduledCustomer

Pickup Time

ActualCustomer

Pickup TimeHoursEarly

HoursPastDue

Ranger 12 0 + 8 = 8 20 10 10 2 —

Explorer 10 8 + 6 = 14 24 12 14 — 2

Econoline 150 3 14 + 3 = 17 20 18 18 1 —

Bronco 1 17 + 15 = 32 33 20 32 — 12

Thunderbird 0 32 + 12 = 44 44 22 44 — 22

Active Model J.1 in myomlab providesadditional insight on the use of single-dimension rules.


b. Under the SPT rule, the sequence starts with the engine block that has the shortest processing time,the Econoline 150, and it ends with the engine block that has the longest processing time, the Bronco.The sequence, along with the flow times, early hours, and past due hours, is contained in the follow-ing table:

The performance measures are

DECISION POINT

The EDD rule is better than the SPT rule with respect to average past due (keeping promises to customers),but worse with respect to average flow time for the set of jobs in this example. Management’s choicedepends on which performance measure it values the most. More experimentation should be conductedbefore a final choice is made.

Average hours past due =

0 + 0 + 7 + 7 + 245

= 7.6 hrs

Average hours early =

15 + 3 + 0 + 0 + 05

= 3.6 hrs

Average flow time =

6 + 19 + 29 + 29 + 455

= 25.6 hrs

Engine BlockSequence

HoursSince Order

ArrivedBeginWork

ProcessingTime (hr)

FinishTime (hr)

Flow Time(hr)

ScheduledCustomer

Pickup Time

ActualCustomer

Pickup TimeHoursEarly

HoursPastDue

Econoline 150 3 0 + 3 = 3 6 18 18 15 —

Explorer 10 3 + 6 = 9 19 12 12 3 —

Ranger 12 9 + 8 = 17 29 10 17 — 7

Thunderbird 0 17 + 12 = 29 29 22 29 — 7

Bronco 1 29 + 15 = 44 45 20 44 — 24

As the solution of Example J.1 shows, the EDD schedule gave better customer service, asmeasured by the average hours past due, and a lower maximum hours past due (22 versus 24).However, the SPT schedule provided a lower average flow time. In general, the SPT priority rulewill push most jobs through the system to completion more quickly than will the other rules.Speed can be an advantage—but only if jobs can be delivered sooner than promised and revenuecollected earlier. If they cannot, the completed job must stay in finished inventory. Consequently,the priority rule chosen can help or hinder the firm in meeting its competitive priorities.

Researchers have studied the implications of the single-dimension rules for variousperformance measures. In most of these studies, all jobs were considered to be independent(in contrast to the parent-component dependencies in MRP environments), and theassumption was made that sufficient capacity generally was available. These studies foundthat the EDD rule performs well with respect to the percentage of jobs past due and the vari-ance of hours past due. For any set of jobs to be processed at a single workstation, it mini-mizes the maximum of the past due hours of any job in the set. The EDD rule is popular withfirms that are sensitive to achieving due dates, which usually are the basis for setting priori-ties using MRP systems.

Often referred to as the world champion, the SPT rule tends to minimize the mean flowtime (assuming time since arrival is 0 for all jobs) and the percentage of jobs past due. It alsotends to maximize shop utilization. For the single-workstation case, the SPT rule always willprovide the lowest mean finish time. However, it could increase total inventory because ittends to push all work to the finished state. In addition, it tends to produce a large variancein past due hours because the larger jobs might have to wait a long time for processing. Also,it provides no opportunity to adjust schedules when due dates change. The advantage of thisrule over others diminishes as the load on the shop increases.

Finally, though the FCFS rule is considered fair to the jobs (or customers), it performspoorly with respect to all performance measures. This result is to be expected because FCFSdoes not acknowledge any job (or customer) characteristics. However, FCFS usually is theonly acceptable choice for service processes where the customer is present and demand lev-eling options such as appointments or reservations are not used.


Multiple-Dimension Rules Priority rules, such as CR and S/RO, incorporate informationabout the remaining workstations at which the job must be processed, in addition to theprocessing time at the present workstation or the due date considered by single-dimensionrules. We call these rules multiple-dimension rules because they apply to more than oneaspect of the job. Example J.2 demonstrates their use for sequencing jobs.

Job

Processing Time at EngineLathe (hours)

TimeRemainingUntil Due

Date (days)

Number ofOperationsRemaining

Shop TimeRemaining

(days) CR S/RO

1 2.3 15 10 6.1 2.46 0.89

2 10.5 10 2 7.8 1.28 1.10

3 6.2 20 12 14.5 1.38 0.46

4 15.6 8 5 10.2 0.78 0.44-

Priority Rule Summary

FCFS SPT EDD CR S/RO

Average flow time 17.175 16.100 26.175 27.150 24.025

Average early time 3.425 6.050 0 0 0

Average past due 7.350 8.900 12.925 13.900 10.775

multiple-dimension rules

A set of rules that apply to more than oneaspect of a job.

Sequencing with the CR and S/RO RulesEXAMPLE J.2

The first five columns of the following table contain information about a set of four jobs that just arrived (end ofhour 0 or beginning of hour 1) at an engine lathe. They are the only ones now waiting to be processed. Severaloperations, including the one at the engine lathe, remain to be done on each job. Determine the schedule by using(a) the CR rule and (b) the S/RO rule. Compare these schedules to those generated by FCFS, SPT, and EDD. Tutor J.2 in myomlab provides a

new example to practice the CR andS/RO rules.

SOLUTION

a. Using CR to schedule the machine, we divide the time remaining until the due date by the shop timeremaining to get the priority index for each job. For job 1,

By arranging the jobs in sequence with the lowest critical ratio first, we determine that the sequenceof jobs to be processed by the engine lathe is 4, 2, 3, and finally 1, assuming that no other jobs arrivein the meantime.

b. Using S/RO, we divide the difference between the time remaining until the due date and the shop timeremaining by the number of remaining operations. For job 1,

Arranging the jobs by starting with the lowest S/RO yields a 4, 3, 1, 2 sequence of jobs.

DECISION POINT

Note that the application of the two priority rules gives two different schedules. Moreover, the SPT sequence,based on processing times (measured in hours) at the engine lathe only, is 1, 3, 2, and 4. No preference is givento job 4 in the SPT schedule, even though it may not be finished by its due date. The EDD sequence is 4, 2, 1,and 3. For illustration purposes, we assume that the FCFS sequence is 1, 2, 3, and 4. All four jobs arrived at theworkstation at the end of hour 0, so the finish times and flow times are identical for all five rules. The followingtable shows the comparative performance of the five priority sequencing rules at the engine lathe:

S/RO =

Time remaining until the due date - Shop time remaining

Number of operations remaining=

15 - 6.110

= 0.89

CR =

Time remaining until the due date

Shop time remaining=

156.1

= 2.46


The S/RO rule is better than the EDD rule and the CR rule, but it is much worse than the SPT rule and theFCFS rule for this example. However, EDD, CR, and S/RO all have the advantage of allowing schedulechanges when due dates change. These results cannot be generalized to other situations because only fourjobs are being processed.

Johnson’s rule

A procedure that minimizes makespanwhen scheduling a group of jobs on twoworkstations.

Research studies have shown that S/RO is better than EDD with respect to the per-centage of jobs past due but worse than SPT and EDD with respect to average flow times.These studies also indicate that CR results in longer flow times than SPT, but CR alsoresults in less variance in the distribution of past due hours. Consequently, even thoughthe use of the multiple-dimension rules requires more information, no choice is clearlybest. Each rule should be tested in the environment for which it is intended.

Scheduling Jobs for Multiple WorkstationsPriority sequencing rules can be used to schedule more than one operation. Each operationis treated independently. When a workstation becomes idle, the priority rule is applied tothe jobs waiting for that operation, and the job with the highest priority is selected. Whenthat operation is finished, the job is moved to the next operation in its routing, where itwaits until it again has the highest priority. At any workstation, the jobs in the waiting linechange over a period of time, so the choice of a priority rule can make quite a difference inthe processing sequence. Schedules can be evaluated with the performance measuresalready discussed.

Identifying the best priority rule to use at a particular operation in a process is a com-plex problem because the output from one operation becomes the input to another. The pri-ority rule at a workstation determines the sequence of work the workstation will perform,which in turn determines the arrival of work at the next workstation downstream. Computersimulation models are effective tools to determine which priority rules work best in a givensituation. Once the current process is modeled, the analyst can make changes to the prior-ity rules at various operations and measure the impact on performance measures, such aspast due, flow time, and utilization.

Scheduling Jobs for a Two-Station Flow ShopSuppose that a flow shop has several jobs ready for processing at two workstations and that the routings of all jobs are identical. In the scheduling of two or more workstationsin a flow shop, the makespan varies according to the sequence chosen. Determin-ing a production sequence for a group of jobs to minimize the makespan has two advantages:

1. The group of jobs is completed in minimum time.

2. The utilization of the two-station flow shop is maximized. Utilizing the first workstationcontinuously until it processes the last job minimizes the idle time on the secondworkstation.

Johnson’s rule is a procedure that minimizes makespan when scheduling a group ofjobs on two workstations. S. M. Johnson showed that the sequence of jobs at the two stationsshould be identical and that the priority assigned to a job should, therefore, be the same atboth. The procedure is based on the assumption of a known set of jobs, each with a knownprocessing time and available to begin processing on the first workstation. The procedure isas follows.

Step 1. Scan the processing times at each workstation and find the shortest processingtime among the jobs not yet scheduled. If two or more jobs are tied, choose one jobarbitrarily.

Step 2. If the shortest processing time is on workstation 1, schedule the corresponding jobas early as possible. If the shortest processing time is on workstation 2, schedule the corre-sponding job as late as possible.

Step 3. Eliminate the last job scheduled from further consideration. Repeat steps 1 and2 until all jobs have been scheduled.


Establishing a Job Sequence

Iteration Job Sequence Comments

1 M3 The shortest processing time is 3 hours for M3 at workstation 2. Therefore, M3 isscheduled as late as possible.

2 M2 M3 Eliminate M3 from the table of estimated times. The next shortest processing timeis 4 hours for M2 at workstation 1. M2 is therefore scheduled first.

3 M2 M5 M3 Eliminate M2 from the table. The next shortest processing time is 8 hours for M5at workstation 2. Therefore, M5 is scheduled as late as possible.

4 M2 M1 M5 M3 Eliminate M5 from the table. The next shortest processing time is 12 hours for M1at workstation 1. M1 is scheduled as early as possible.

5 M2 M1 M4 M5 M3 The last motor to be scheduled is M4. It is placed in the last remaining position, inthe middle of the schedule.

� FIGURE J.2Gantt Chart for the Morris MachineCompany Repair Schedule

0 5 10 15 20 25 30 35 40 45 50 55 60 65

1 M2 (4)

M1 (12)

M4 (15)

M5 (10)

M3 (5)

Idle—available for further work

2 Idle IdleM2 (5)

M1 (22)

M4 (16)

M5 (8)

M3 (3)

Workstation

Hour

Time (hr)

Motor Workstation 1 Workstation 2

M1 12 22

M2 4 5

M3 5 3

M4 15 16

M5 10 8

Scheduling a Group of Jobs on Two WorkstationsEXAMPLE J.3

The Morris Machine Company just received an order to refurbish five motors for materials handling equipmentthat were damaged in a fire. The motors have been delivered and are available for processing. The motors will berepaired at two workstations in the following manner.

Workstation 1: Dismantle the motor and clean the parts.Workstation 2: Replace the parts as necessary, test the motor, and make adjustments.

The customer’s shop will be inoperable until all the motors have been repaired, so the plant manager isinterested in developing a schedule that minimizes the makespan and has authorized around-the-clock operationsuntil the motors have been repaired. The estimated time to repair each motor is shown in the following table:

Tutor J.3 in myomlab provides a newexample to practice Johnson’s rule.

SOLUTION

The logic for the optimal sequence is shown in the following table:

DECISION POINT

No other sequence of jobs will produce a shorter makespan. To determine the makespan, we can draw a Ganttchart, as shown in Figure J.2. In this case, refurbishing and reinstalling all five motors will take 65 hours. Thisschedule minimizes the idle time of workstation 2 and gives the fastest repair time for all five motors. Note thatthe schedule recognizes that a job cannot begin at workstation 2 until it has been completed at workstation 1.


labor-limited environment

An environment in which the resourceconstraint is the amount of laboravailable, not the number of machines orworkstations.

Labor-Limited EnvironmentsThus far, we have assumed that a job never has to wait for lack of a worker. The limitingresource has been the number of machines or workstations available. More typical, however,is a labor-limited environment in which the resource constraint is the amount of laboravailable, not the number of machines or workstations. In this case, workers are trained towork on a variety of machines or tasks to increase the flexibility of operations.

In a labor-limited environment, the scheduler not only must decide which job toprocess next at a particular workstation but also must assign workers to their next worksta-tions. The scheduler can use priority rules to make these decisions, as we used them toschedule engine blocks in Example J.1. In labor-limited environments, the labor-assignmentpolicies, as well as the priority sequencing rules, affect performance. The following examplesprovide some labor-assignment rules.

� Assign personnel to the workstation with the job that has been in the system longest.

� Assign personnel to the workstation with the most jobs waiting for processing.

� Assign personnel to the workstation with the largest standard work content.

� Assign personnel to the workstation with the job that has the earliest due date.

The manufacturing scheduling process is a key element of an integrated supply chain.Advanced planning and scheduling (APS) systems attempt to link the scheduling process todemand data and forecasts, supply chain facility and inventory decisions, and the capabilityof suppliers so that the entire chain can operate as efficiently as possible. A firm’s ability tochange its schedules quickly and still keep the supply chain flowing smoothly provides acompetitive edge.

myomlab and the Companion Website at www.pearsonhighered.com contain many tools, activities, and resources designed forthis supplement.

Internet Resources

Key Equations1. Performance measures:

Flow time = Finish time + Time since the job arrived at the workstation

Past due = Time by which a job missed its due date

Makespan = Time of completion of last job Starting time of the first job

Total inventory = Scheduled receipts for all items + On-hand inventories of all items

2. Critical ratio:

3. Slack per remaining operations:

S/RO =

(Due date - Today’s date) - Total shop time remaining


CR =


Total shop time remaining

-


The Neptune’s Den Machine Shop specializes in overhauling outboard marine engines.Some engines require replacement of broken parts, whereas others need a complete over-haul. Currently, five engines with varying problems are awaiting service. The best estimatesfor the labor times involved and the promise dates (in number of days from today) are shownin the following table. Customers usually do not pick up their engines early.

Repair Sequence

Days SinceOrder

ArrivedProcessing

TimeFinishTime

FlowTime

PromiseDate

ActualPickupDate

DaysEarly

DaysPast Due

50-hp Evinrude 4 5 5 9 8 8 3 —

75-hp Nautique 15 3 8 23 10 10 2 —

100-hp Mercury 8 10 18 26 12 18 — 6

7-hp Johnson 6 4 22 28 15 22 — 7

50-hp Honda 1 1 23 24 20 23 — 3

Total 110

EngineTime Since Order

Arrived (days)Processing Time,

Including Setup (days)Promise Date (days

from now)

50-hp Evinrude 4 5 8

7-hp Johnson 6 4 15

100-hp Mercury 8 10 12

50-hp Honda 1 1 20

75-hp Nautique 15 3 10

Repair Sequence

Days SinceOrder

ArrivedProcessing

TimeFinishTime

FlowTime

PromiseDate

ActualPickupDate

DaysEarly

DaysPast Due

50-hp Honda 1 1 1 2 20 20 19 —

75-hp Nautique 15 3 4 19 10 10 6 —

7-hp Johnson 6 4 8 14 15 15 7 —

50-hp Evinrude 4 5 13 17 8 13 — 5

100-hp Mercury 8 10 23 31 12 23 — 11

Total 83

Solved Problem 1

a. Develop separate schedules by using the SPT and EDD rules.

b. Compare the two schedules on the basis of average flow time, percentage of past duejobs, and maximum past due days for any engine.

SOLUTION

a. Using the SPT rule, we obtain the following schedule:

Using the EDD we obtain this schedule:

b. Performance measures are as follows:

Average flow time is 16.6 (or 83/5) days for SPT and 22.0 (or 110/5) days for EDD. Thepercentage of past due jobs is 40 percent (2/5) for SPT and 60 percent (3/5) for EDD.For this set of jobs, the EDD schedule minimizes the maximum days past due but hasa greater flow time and causes more jobs to be past due.


Current OrderProcessing Time (hr)

Due Date (day)

RemainingOperations

Shop TimeRemaining (days)

A101 10 162 10 9

B272 7 158 9 6

C105 15 152 1 1

D707 4 170 8 18

E555 8 154 5 8

The following data were reported by the shop floor control system for order processing at theedge grinder. The current date is day 150. The number of remaining operations and the totalwork remaining include the operation at the edge grinder. All orders are available for pro-cessing, and none have been started yet. Assume the jobs were available for processing atthe same time.

a. Specify the priorities for each job if the shop floor control system uses slack perremaining operations (S/RO) or critical ratio (CR).

b. For each priority rule, calculate the average flow time per job at the edge grinder.

SOLUTION

a. We specify the priorities for each job using the two sequencing rules.

The sequence of production for S/RO is shown in the preceding brackets.

The sequence of production for CR is shown in the preceding brackets.

b. We are sequencing a set of jobs at a single machine, so each job’s finish time equals thefinish time of the job just prior to it in sequence plus its own processing time. Further,

C105:CR =

152 - 1501

= 2.00 [5]

A101:CR =

162 - 1509

= 1.33 [4]

B272:CR =

158 - 1506

= 1.33 [3]

D707:CR =

170 - 15018

= 1.11 [2]

E555:CR =

154 - 1508

= 0.50 [1]

CR =


Shop time remaining

C105:S/RO =

(152 - 150) - 1

1= 1.00 [5]

A101:S/RO =

(162 - 150) - 9

10= 0.30 [4]

D707:S/RO =

(170 - 150) - 18

8= 0.25 [3]

B272:S/RO =

(158 - 150) - 6

9= 0.22 [2]

E555:S/RO =

(154 - 150) - 8

5= - 0.80 [1]

S/RO =

(Due date - Today’s date) - Shop time remaining


Solved Problem 2


all jobs were available for processing at the same time, so each job’s finish time equalsits flow time. Consequently, the average flow times at this single machine are

In this example, the average flow time per job is lower for the CR rule, which is notalways the case. For example, the critical ratios for B272 and A101 are tied at 1.33. If wearbitrarily assigned A101 before B272, the average flow time would increase to

.(8 + 12 + 22 + 29 + 44)/5 = 23.0 hours

CR: 8 + 12 + 19 + 29 + 44

5= 22.4 hours

S/RO: 8 + 15 + 19 + 29 + 44

5= 23.30 hours

The Rocky Mountain Arsenal, formerly a chemical warfare manufacturing site, is said to beone of the most polluted locations in the United States. Cleanup of chemical waste storagebasins will involve two operations.

Operation 1: Drain and dredge basin.

Operation 2: Incinerate materials.

Management estimates that each operation will require the following amounts of time (in days):

Solved Problem 3

Storage Basin

A B C D E F G H I J

Dredge 3 4 3 6 1 3 2 1 8 4

Incinerate 1 4 2 1 2 6 4 1 2 8

1. Select basin E first (tied with basin H);put it at the front.

E — — — — — — — — —

2. Select basin H next; put it toward the front.

E H — — — — — — — —

3. Select basin A next (tied with basin D); put it at the end.

E H — — — — — — — A

4. Put basin D toward the end. E H — — — — — — D A

5. Put basin G toward the front. E H G — — — — — D A

6. Put basin C toward the end. E H G — — — — C D A

7. Put basin I toward the end. E H G — — — I C D A

8. Put basin F toward the front. E H G F — — I C D A

9. Put basin B toward the front. E H G F B — I C D A

10. Put basin J in the remaining space. E H G F B J I C D A

Management’s objective is to minimize the makespan of the cleanup operations. Allstorage basins are available for processing right now. First, find a schedule that minimizesthe makespan. Then calculate the average flow time of a storage basin through the two oper-ations. What is the total elapsed time for cleaning all 10 basins? Display the schedule in aGantt machine chart.

SOLUTION

We can use Johnson’s rule to find the schedule that minimizes the total makespan. Fourjobs are tied for the shortest process time: A, D, E, and H. E and H are tied for first place,while A and D are tied for last place. We arbitrarily choose to start with basin E, the firston the list for the drain and dredge operation. The 10 steps used to arrive at a sequenceare as follows:


Storage basin

EDredge H G F B J I C D A

Incinerate E H G F B J I C DD A

FIGURE J.3 �

OrderTime Since Order

Arrived (hr)Estimated

ProcessingTime (hr)Due Date

(hr from now)

1 12 10 12

2 10 3 8

3 7 15 18

4 3 9 20

5 1 7 21

Several optimal solutions are available to this problem because of the ties at the start ofthe scheduling procedure. However, all have the same makespan. The schedule would be asfollows:

The makespan is 36 days. The average flow time is the sum of incineration finish timesdivided by 10, or 200/10 = 20 days. The Gantt machine chart for this schedule is given inFigure J.3.

Operation 1 Operation 2

Basin Start Finish Start Finish

E 0 1 1 3

H 1 2 3 4

G 2 4 4 8

F 4 7 8 14

B 7 11 14 18

J 11 15 18 26

I 15 23 26 28

C 23 26 28 30

D 26 32 32 33

A 32 35 35 36

Total 200

Discussion Question1. Suppose that two alternative approaches for determining

workstation schedules are available. One is an optimizingapproach that can be run once a week on the computer.The other approach utilizes priority sequencing rules to

determine the schedule as it evolves. Discuss the advan-tages and disadvantages of each approach and the condi-tions under which each approach is likely to be better.

ProblemsSoftware, such as OM Explorer, Active Models, and POM forWindows, is available in myomlab. Check with your instructoron how best to use it. In many cases, the instructor wants you tounderstand how to do the calculations by hand. At most, thesoftware provides a check on your calculations. When calcula-tions are particularly complex and the goal is interpreting theresults in making decisions, the software replaces entirely themanual calculations.

1. The Hickory Company manufactures wooden desks.Management schedules overtime every weekend toreduce the backlog on the most popular models. Theautomatic routing machine is used to cut certain types of

edges on the desktops. The following orders need to bescheduled for the routing machine:


� FIGURE J.4

A Job 1

Job 2

Job 3

Idle

0 1 2 3 4 5 6 7 8 9

B Idle Job 1

Job 2

Job 3

Machine

The due dates reflect the need for the order to be at its nextoperation.

a. Develop separate schedules by using the FCFS, SPT,and EDD rules.

b. Compare the schedules on the basis of average flowtime, the average early time, and average past duehours for any order.

c. Comment on the performance of the two rules relativeto these measures.

2. The drill press is a bottleneck operation. Currently, fivejobs are waiting to be processed. Following are the avail-able operations data. Assume that the number of remain-ing operations and the shop time remaining include theprocessing at the drill press.

a. Specify the priority for each job if the shop floor con-trol system uses each of the following priority rules:SPT, S/RO, EDD, and CR.

b. For each priority rule, calculate the average flow timeper job at the drill press.

c. Which of these priority rules would work best for pri-ority planning with an MRP system? Why?

Job

Time SinceOrder

Arrived (hr)ProcessingTime (hr)

Time toDue Date

(wk)OperationsRemaining

Shop TimeRemaining

(wk)

AA 24 4 10 3 4

BB 16 8 16 4 6

CC 14 13 21 10 9

DD 12 6 23 3 12

EE 10 2 12 5 3

TABLE J.1 MANUFACTURING DATA

JobRelease

TimeLotSize

ProcessingTime

(hr/unit)

SetupTime(hr)

DueDate

1 9:00 A.M.Monday

50 0.06 4 9:00 P.M.Monday

2 10:00 A.M.Monday

120 0.05 3 10:00 P.M.Monday

3 11:00 A.M.Monday

260 0.03 5 11:00 P.M.Monday

4 12:00 P.M.Monday

200 0.04 2 2:00 A.M.Tuesday

3. The machine shop at Bycraft Enterprises operates 24 hoursa day and uses a numerically controlled (NC) weldingmachine. The load on the machine is monitored, and nomore than 24 hours of work is released to the weldingoperators in one day. The data for a typical set of jobs areshown in Table J.1. Management has been investigating

scheduling procedures that would reduce inventory andincrease customer service in the shop. Assume that at9:00 A.M. on Monday the NC welding machine was idle.Also assume that job “arrival times” are the “releasetimes” to the workstation.

a. Develop schedules for SPT and EDD priority rules,and draw a Gantt machine chart for each schedule.

b. For each schedule in part (a), calculate the averageflow time per job and the average past due hours per job.

4. Refer to the Gantt machine chart in Figure J.4.

a. Suppose that a routing requirement is that each jobmust be processed on machine A first. Can themakespan be improved? If so, draw a Gantt chart withthe improved schedule. If not, state why.

b. Suppose that the machine sequence has no routingrestriction; in other words, jobs can be processed inany sequence on the machines. Can the makespan inthe chart be improved in this case? If so, draw a Ganttchart with your schedule. If not, state why.

5. A manufacturer of sails for small boats has a group ofcustom sails awaiting the last two processing operationsbefore the sails are sent to the customers. Operation 1must be performed before operation 2, and the jobs havedifferent time requirements for each operation. Thehours required are as follows:

Job

1 2 3 4 5 6 7 8 9 10

Operation 1 1 5 8 3 9 4 7 2 4 9

Operation 2 8 3 1 2 8 6 7 2 4 1

a. Use Johnson’s rule to determine the optimal sequence.

b. Draw a Gantt chart for each operation.

6. McGee Parts Company is under tremendous pressure tocomplete a government contract for six orders in 31 work-ing days. The orders are for spare parts for highway main-tenance equipment. According to the government con-tract, a late penalty of $1,000 is imposed each day theorder is late. Owing to a nationwide increase in highwayconstruction, McGee Parts has received many orders for


spare parts replacement and the shop has beenextremely busy. To complete the government contract,the parts must be deburred and heat treated. The pro-duction control manager has suggested the followingschedule:

Debur Heat Treat

Job Start Finish Start Finish

1 0 2 2 8

2 2 5 8 13

3 5 12 13 17

4 12 15 17 25

5 15 16 25 30

6 16 24 30 32

a. Use Johnson’s rule to determine the optimal sequence.

b. Draw a Gantt chart for each operation.

7. Carolyn Roberts is the operations manager of themachine shop of Reliable Manufacturing. She has toschedule eight jobs that are to be sent to final assemblyfor an important customer order. Currently, all eight jobsare in department 12 and must be routed to department 22next. All jobs arrived at the same time. Jason Mangano,supervisor for department 12, is concerned about keeping

Job

1 2 3 4 5 6 7 8

Department 12 2 4 7 5 4 10 8 2

Department 22 3 6 3 8 2 6 6 5

a. Determine a schedule for the operation in eachdepartment. Use SPT for department 12 and the samesequence for department 22. What is the average flowtime for department 12? What is the makespanthrough both departments? What is the total numberof job-days spent in the system?

b. Find a schedule that will minimize the makespanthrough both departments, and then calculate theaverage flow time for department 12. What is the totalnumber of job-days spent in the system?

c. Discuss the trade-offs represented by these two sched-ules. What implications do they have for centralizedscheduling?

his inventory low and is adamant about processing thejobs through his department on the basis of shortest pro-cessing time. Pat Mooney, supervisor for department 22,pointed out that if Mangano were more flexible the orderscould be finished and shipped earlier. The processingtimes (in days) for each job in each department follow:

ADVANCED PROBLEMS

8. The repair manager at Standard Components needs todevelop a schedule for repairing eight Dell PCs. Each jobrequires analysis using the same diagnostic system.Furthermore, each job will require additional processingafter the diagnostic evaluation. The manager does notexpect any rescheduling delays, and the jobs are to movedirectly to the next process after the diagnostic work hasbeen completed. The manager has collected the followingprocessing time and scheduling data for each repair job:

a. Compare the relative performance of the FCFS, SPT,EDD, S/RO, and CR rules in terms of the percent ofjobs past due, average days past due, and maximumdays of past due. (Hint: The time since an order was

Job

Time SinceOrder Arrived

(days)

ProcessingTime

(days)

Due Date(days

from now)

Shop TimeRemaining

(days)OperationsRemaining

1 10 1.25 6 2.5 5

2 9 2.75 5 3.5 7

3 7 2.50 7 4.0 9

4 6 3.00 6 4.5 12

5 5 2.50 5 3.0 8

6 4 1.75 8 2.5 6

7 3 2.25 7 3.0 9

8 1 2.00 5 2.5 3

placed is needed just to establish the sequence for theFCFS rule, because all performance measures dealwith past due statistics.)

b. Discuss the selection of one of the rules for this com-pany. What criteria do you consider most important inthe selection of a rule in this situation?

9. Penultimate Support Systems makes fairly good speakerand equipment support stands for music groups. Theassembly process involves two operations: (1) fabrication,or cutting aluminum tubing to the correct lengths, and (2) assembly, with purchased fasteners and injection-molded plastic parts. Setup time for assembly is negligible.Fabrication setup time and run time per unit, assembly runtime per unit, and the production schedule for next weekfollow. All jobs arrived at the same time. Organize the workto minimize makespan, and create a Gantt chart. Can thiswork be accomplished within two 40-hour shifts?

Fabrication Assembly

Model Quantity Setup (hr)Run Time(hr/unit)

Run Time(hr/unit)

A 200 2 0.050 0.04

B 300 3 0.070 0.10

C 100 1 0.050 0.12

D 250 2 0.064 0.60


Job

1 2 3 4 5 6 7 8

Machine 1 2 5 2 3 1 2 4 2

Machine 2 4 1 3 5 5 6 2 1

Machine 3 6 4 5 2 3 2 6 2

10. Eight jobs must be processed on three machines in thesequence M1, M2, and M3. The processing times (inhours) are as follows:

Machine M2 is a bottleneck, and management wants tomaximize its use. Consequently, the schedule for theeight jobs, through the three machines, was based on theSPT rule on M2. The proposed schedule is 2, 8, 7, 3, 1, 4,5, and 6.

a. It is now 4:00 P.M. on Monday. Suppose that processingon M2 is to begin at 7:00 A.M. on Tuesday. Use the pro-posed schedule to determine the schedules for M1 andM3 so that job 2 begins processing on M2 at 7:00 A.M. onTuesday. Draw Gantt charts for M1, M2, and M3. Whatis the makespan for the eight jobs?

b. Find a schedule that utilizes M2 better and yields ashorter makespan.

11. The last few steps of a production process require twooperations. Some jobs require processing on M1 before

Processing Time (hr)

Job M1 M2 M3 Due Date (hr from now)

1 6 — 4 13

2 2 — 1 18

3 4 — 7 22

4 5 — 3 16

5 7 — 4 30

6 3 — 1 29

7 — 4 6 42

8 — 2 10 31

9 — 6 9 48

10 — 8 2 40

processing on M3. Other jobs require processing on M2before M3. Currently, six jobs are waiting at M1 and fourjobs are waiting at M2. The following data have been sup-plied by the shop floor control system:

a. Schedule this shop by using the following rules: SPT,EDD, S/RO, and CR.

b. Discuss the operating implications of each of theschedules you developed in part (a). Assume all jobsarrived at the same time.

Active Model ExerciseThis Active Model appears in myomlab. It allows you to evalu-ate the application of single-dimension priority rules forscheduling jobs at one workstation.

QUESTIONS

1. Which rule minimizes the average job flow time in thesystem for this example?

2. Use the scroll bars to change the five processing timesand the five due dates. Does the same rule always mini-mize the average flow time and average past due?

3. Which rule minimizes the average hours past due for thisexample?

4. Use the scroll bar to change the processing time for theThunderbird and to modify the due date for theThunderbird. Does the same rule always minimize theaverage hours past due?

5. Which rule minimizes the average hours early for thisexample?

6. Use the scroll bar to change the processing time for theEconoline and to modify the due date for the Econoline.Does the same rule always minimize the average hourspast due?


Baker, K. R. Elements of Sequencing and Scheduling. Hanover,NH: Baker Press, 2002.

Hartvigsen, David. SimQuick: Process Simulation with Excel,2nd ed. Upper Saddle River, NJ: Prentice Hall, 2004.

Johnson, S. M. “Optimal Two Stage and Three Stage ProductionSchedules with Setup Times Included.” Naval LogisticsQuarterly, vol. 1, no. 1 (1954), pp. 61–68.

Kiran, Ali S., and Thomas H. Willingham. “Simulation: Help forYour Scheduling Problems.” APICS—The PerformanceAdvantage (August 1992), pp. 26–28.

LaForge, R. Lawrence, and Christopher W. Craighead.“Computer-Based Scheduling in Manufacturing Firms: SomeIndicators of Successful Practice.” Production and InventoryManagement Journal (First Quarter 2000), pp. 29–34.

Metters, Richard, and Vincente Vargas. “A Comparison ofProduction Scheduling Policies on Costs, Service Levels,

Job Shop Scheduling Using Data from Example J.1

Selected Referencesand Schedule Changes.” Production and OperationsManagement, vol. 17, no. 3 (1999), pp. 76–91.

Pinedo, Michael. Scheduling: Theory, Algorithms, and Systems,2nd ed. Upper Saddle River, NJ: Prentice Hall, 2002.

Pinedo, M., and X. Chao. Operations Scheduling withApplications in Manufacturing and Services. Boston:McGraw-Hill/Irwin, 1998.

Suresh, V., and D. Chaudhuri. “Dynamic Scheduling-A Surveyof Research.” International Journal of ProductionEconomics, vol. 32 (1993), pp. 52–63.

Vollmann, Thomas E., William Berry, D. Clay Whybark, andRobert Jacobs. Manufacturing Planning and ControlSystems for Supply Chain Management, 5th ed. New York:McGraw-Hill/Irwin, 2005.

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